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Let <i>G</i> be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on <i>G</i> to the tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>G</mi></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>G</mi></mrow></semantics></math></inline-formula>, and we express in different cases the contravariant Levi-Civita connection and curvature of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>G</mi></mrow></semantics></math></inline-formula> in terms of the contravariant Levi-Civita connection and the curvature of <i>G</i>. We prove that the space of differential forms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>Ω</mo><mo>*</mo></msup><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> on <i>G</i> is a differential graded Poisson algebra if, and only if, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>Ω</mo><mo>*</mo></msup><mrow><mo stretchy="false">(</mo><mi>T</mi><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is a differential graded Poisson algebra. Moreover, we show that <i>G</i> is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>G</mi></mrow></semantics></math></inline-formula> is also a pseudo-Riemannian Poisson–Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson–Lie groups are given.